Presentation at SIAM Conference on Applied Algebraic Geometry (AG21), Aug. 2021. Abstract. The study of phase transitions using data-driven approaches is challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has recently achieved success in detecting structural transitions in material science, such as the glass--liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We thus propose a general framework, termed “topological persistence machine," to construct the shape of data from correlations in states so that we can subsequently decipher phase transitions via qualitative changes in the shape. Our framework enables an effective and unified approach in phase transition analysis without having prior knowledge about phases or requiring the investigation of the system with large size. We demonstrate the efficacy of the approach in terms of detecting the Berezinskii--Kosterlitz--Thouless phase transition in the classical XY model and quantum phase transitions in the transverse Ising and Bose--Hubbard models. Interestingly, while these phase transitions have proven to be notoriously difficult to analyze using traditional methods, they can be characterized through our framework without requiring prior knowledge of the phases. Our approach is thus expected to be widely applicable and will provide the prospective with practical interests in exploring the phases of experimental physical systems.

1. Topological Persistence Machine
of Phase Transition
The University ofTokyo
Tran Quoc Hoan
(join work with M. Chen andY. Hasegawa)
MS:Applications of Persistent Homology to PhaseTransitions

2. Motivation
Data-driven approach in physical system with rich geometric and
topological structure
Quantum Quench
Dynamics
Spins Configuration
Interaction
Network
“It’s black but not a black box.”
AlgebraicTopological Machine
Topological Features
2/30
Topological Data Analysis = Model the Shape of Data

3. Persistent Homology
Main idea: vary a proximity parameter and track the appearance and
disappearance of features
Barcodes
Persistence
Diagram
Apply Homology
S. Barannikov+(1994),H. Edelsbrunner+(2002),
A. Zomorodian and G. Carlsson (2004),…
Significant features
persist
Filtration
3/30

4. Persistent Homology
Mathematical encoding: model the filtration as an increasing sequence of
simplicial complexes
S. Barannikov+(1994),H. Edelsbrunner+(2002),
A. Zomorodian and G. Carlsson (2004),…
⊆ ⊆ ⊆
𝜎1
𝜎1
𝜎2 𝜎3
𝜎4
𝜎1
𝜎2 𝜎3
𝜎4 𝜎5
𝜎6
𝜎1
𝜎2 𝜎3
𝜎4 𝜎5
𝜎6
𝜎7
𝐾1 𝐾2 𝐾3 𝐾4
𝐶𝑝 𝐾 = σ𝑖 𝑎𝑖𝜎𝑖 ∣ 𝜎𝑖 ⊂ 𝐾, dim 𝜎𝑖 = 𝑝, 𝑎𝑖 ∈ ℤ2
◼ TheVietoris-Rips complex VR 𝑃, 𝜀
𝑉𝑅 𝑃, 𝜀 = { 𝑖0, … , 𝑖𝑘 ∣ 𝑑 𝑥𝑖𝑎
, 𝑥𝑖𝑏
≤ 2𝜀 ∀0 ≤ 𝑎, 𝑏 ≤ 𝑘}
1 2
3
𝐵1 𝐵2
𝐵3
◼ Filtration
𝜀1 < 𝜀2 < ⋯ < 𝜀𝑇
→ 𝑉𝑅 𝑃, 𝜀1 ⊆ 𝑉𝑅 𝑃, 𝜀2 ⊆ ⋯ ⊆ 𝑉𝑅 𝑃, 𝜀𝑇 4/30

5. Persistent Homology S. Barannikov+(1994),H. Edelsbrunner+(2002),
A. Zomorodian and G. Carlsson (2004),…
Simplicial
complex
Chain
complex
Homology
group
Algebraic Holes
(Geometrical object) (Algebraic object) (Algebraic object)
𝐻p =
Ker(𝜕𝑝)
Im(𝜕𝑝+1)
Image by Yasuaki Hiraoka via
http://www.wpi-
aimr.tohoku.ac.jp/hiraoka_labo/index-english.html
Standard computational time O(M3), M = number of simplices
Nina Otter+, A roadmap for the computation of persistent homology, EPJ Datascience, 2017
Complex K Size of K Theoretical guarantee
Čech 2𝑂 |𝑃| Nerve theorem
Vietoris-Rips (VR) 2𝑂 |𝑃| Approximate Čech complex
Alpha |𝑃|𝑂([𝑑/2]) (𝑁 points in ℝ𝑑) Nerve theorem
Witness 2𝑂 |𝐿|
(subset L) For curves and surfaces in Euclidean space
Graph-induced complex 2𝑂 |𝑄|
(subsample Q) ApproximatesVR complex
Sparsified Čech 𝑂 |𝑃| Approximate Čech complex
SparsifiedVR 𝑂 |𝑃| ApproximatesVR complex
5/30

6. Persistent Homology
Persistent Homology encodes both global and local topology of a
dataset into a computational feature set
[F. C. Motta+, Measures of order for nearly hexagonal lattices, Physica D (2018)] 6/30
Thank to Henry Adam for giving this
example

7. Persistent Homology and Physics of Intelligence
Questions:
◼ Given observations from two groups/phases in a physical system, what
makes them “truly” be different?
◼ Can we know the “key” parameters
underlying the observations?
◼ Can we interpret the phase transition
from features of observations?
◼ Can we predict/infer an unknown phase
transition from limited observations?
Let Persistent
Homology do it
Let Statistical Tools and
Machine Learning do it
+
7/30

8. Persistent Homology and PhaseTransition
◼ Hoan Tran - University ofTokyo, Japan - Topological Persistence Machine of PhaseTransitions
◼ Bart Olsthoorn - Nordic Institute forTheoretical Physics, Sweden - Mapping Complex Phase Diagrams in
Spin Models
◼ Alex Cole - University of Amsterdam, Netherlands - Quantitative and Interpretable Order Parameters
for PhaseTransitions from Persistent Homology
◼ Nick Sale - Swansea University, United Kingdom - Quantitative Analysis of PhaseTransitions using
Persistent Homology
◼ Irene Donato - Nextatlas, Italy - Persistent Homology Analysis of PhaseTransitions
◼ Kouji Kashiwa - Fukuoka Institute of Technology, Japan - Exploring the Phase Structure of QCD Effective
Models with Persistent Homology
◼ Daniel Spitz - Universität Heidelberg, Germany - Universal Dynamics in Quantum Many-Body Systems via
Persistent Homology
◼ Willem Elbers - Durham University, United Kingdom - Topological Signatures of Cosmic Reionization and
the First Galaxies
MS72
MS83
8/30

9. Topological Persistence Machine
(1) Decide observations (2) Embedding (3) Make filtration
(4) Compute diagrams for all observations
(each observation = one dataset)
(5) Point summaries or kernel trick
Q. H.Tran et al., Topological persistence machine of phase
transition, PRE (2021) 9/30

10. Point Summaries of Diagrams
Further compressed features from persistence diagrams
◼ Maximum lifetime 𝒫𝑚𝑎𝑥 𝐷 = max
𝑏,𝑑 ∈𝐷
|𝑑 − 𝑏|
◼ 𝛾-norm 𝒫𝛾 𝐷 = ෍
𝑏,𝑑 ∈𝐷
𝑑 − 𝑏 𝛾
1/𝛾
◼ Normalized entropy
ℰ 𝐷 = −
1
log 𝒫1(𝐷)
෍
𝑏,𝑑 ∈𝐷
|𝑑 − 𝑏|
𝒫1(𝐷)
log
|𝑑 − 𝑏|
𝒫1(𝐷)
(Cohen-Steiner+, 2010)
(Chintakunta+, 2015;Myers+, 2019)
10/30

11. The space of Persistence Diagrams
◼ Not a vector space
◼ Difficult to use in statistical-learning
tasks (e.g., classification, regression)
◼ Cannot define an inner product
KernelTrick for Persistence Diagrams
AVG( , , )
is meaningless.
Idea: map Persistence
Diagrams into a Hilbert space
11/30

12. KernelTrick for Persistence Diagrams
◼ Can define an inner product
◼ Use in (linear) statistical-learning tasks (e.g., SVM)
𝐷1
𝐷2
Ω
Φ𝐷1
Φ𝐷2
, 𝐻𝑏
Feature mapping
Φ 𝐻𝑏 = 𝐿2
(ℝ2
)
∞ −dimensional 𝐿2 space
◼ A map 𝑘: Ω × Ω → ℝ is called kernel if there is a Hilbert
space (𝐻𝑏, ⋅,⋅ ) and a feature map Φ: Ω → 𝐻𝑏 s.t.
Ω × Ω
𝐻b × 𝐻b
ℝ
⋅,⋅
𝑘
(Φ, Φ)
Reininghaus+, 2005; Bubenik+, 2015;
Kusano+, 2016; Chachólski+, 2017;
Adams+, 2017; Carrière+, 2017;
Le+, 2018; Corbet+, 2019;…
𝜙𝐷1
, 𝜙𝐷2 𝐻𝑏
= 𝑘 𝐷1, 𝐷2 for ∀𝐷1, 𝐷2 ∈ Ω
12/30

13. Kernel method
(Corbet+, 2019)The feature map Φ: Ω → 𝐿2
(ℝ2
) is often given by
𝐷 ⟼ ෍
𝒑∈𝐷
𝑤 𝒑 𝑓(⋅, 𝒑)
Peak function
(e.g., Gaussian)
weight
Persistence Scale Space Kernel
(Reininghaus+, 2015)
Persistence Weighted Gaussian
Kernel (Kusano+, 2016)
Persistence Images
(Adam+, 2017)
The kernel is given by the inner product:
𝑘 𝐷1, 𝐷2 = න
ℝ2
Φ𝐷1
𝒑 Φ𝐷2
𝒑 𝑑𝐿2
13/30

14. Kernel method
We can embed Φ𝐷 into a point
ℙ = {𝜌| ‫׬‬ℝ2 𝜌 𝒙 = 1 , 𝜌 𝒙 ≥ 0}
Positive orthant
in the probability simplex
𝜌𝐷 =
1
𝑍
෍
𝒑∈𝐷
𝒩(𝒑, 𝜈𝑰)
{𝒙 = (𝑥1, … , 𝑥𝑛) ∈ ℝ𝑛| σ𝑖
𝑛
𝑥𝑖 = 1, 𝑥𝑖 ≥ 0}
ℎ 𝒙 = 𝑥1, … , 𝑥𝑛 = (𝑦1, … , 𝑦𝑛)
𝕊+ = {𝜒| ‫׬‬
ℝ2 𝜒2 𝒙 = 1 , 𝜒 𝒙 ≥ 0}
Fisher Information Metric
𝑑𝐹(𝜌𝐷1
, 𝜌𝐷2
) = arccos ℎ 𝜌𝐷1
, ℎ 𝜌𝐷2
𝑘 𝐷1, 𝐷2 = exp −𝛼𝑑𝐹(𝜌𝐷1
, 𝜌𝐷2
)
The kernel is given by
Persistence Fisher Kernel
(Le+, 2018)
14/30

15. Applications of Kernel method
◼ Kernel PCA, Kernel SVM
◼ Kernel change point detection with Kernel Fisher Discriminant Ratio (KFDR)
➢ Diagrams along index 𝑠: 𝐷𝑠, 𝑠 = 1, … , 𝑀
➢ For each 𝑠 > 1, two classes are defined by the data before and after 𝒔 and
compute
Ƹ
𝜇1 =
1
𝑠 − 1
෍
𝑖=1
𝑠−1
Φ𝐷𝑖
෠
Σ1 =
1
𝑠 − 1
෍
𝑖=1
𝑠−1
Φ𝐷𝑖
− Ƹ
𝜇1 ⨂ Φ𝐷𝑖
− Ƹ
𝜇1
KFDR𝑀,𝑠,𝛾 =
(𝑠 − 1)(𝑀 − 𝑠 + 1)
𝑀
Ƹ
𝜇2 − Ƹ
𝜇1, ෠
Σ + 𝛾𝐼
−1
Ƹ
𝜇2 − Ƹ
𝜇1
ℋb
while
➢ Find change point as 𝑠𝑐 = max
𝑠>1
KFDR𝑀,𝑠,𝛾 (Kusano+,2015;Tran+, 2019)
෠
Σ2 =
1
𝑀 − 𝑠 + 1
෍
𝑖=𝑠
𝑀
Φ𝐷𝑖
− Ƹ
𝜇2 ⨂ Φ𝐷𝑖
− Ƹ
𝜇2
Ƹ
𝜇2 =
1
𝑀 − 𝑠 + 1
෍
𝑖=𝑠
𝑀
Φ𝐷𝑖
෠
Σ =
𝑠−1
𝑀
෠
Σ1 +
𝑀−𝑠+1
𝑀
෠
Σ2
➢ Change-point regression with Φ𝐷1
, … , Φ𝐷𝑀
Kernel Change-point Analysis
(Harchaoui+, 2009)
15/30

16. Topological Persistence Machine
(1) Decide observations (2) Embedding (3) Make filtration
(4) Compute diagrams for all observations
(each observation = one dataset)
(5) Point summaries or kernel trick
Q. H.Tran et al., Topological persistence machine of phase
transition, PRE (2021) 16/30

17. Case-study: 2D-XY model
(no discontinuity in any
observable such as
magnetization or energy,
infinite order transition)
𝑇
Infinite energy to excite a single vortex,
but thermal fluctuations can create
vortex-antivortex pairs bounding
Low High
Entropically favorable for
vortices to separate
BKT transition
𝑻𝒄
𝑱
≅ 𝟎. 𝟖𝟗
𝛽𝐻 = −
𝐽
𝑘𝐵𝑇
෍
𝑖,𝑗
𝑆𝑖 ⋅ 𝑆𝑗
Image by Matthew Beach via
https://mbeach42.github.io/
17/30

18. Topological defect
𝑤 = 0 𝑤 = 1 𝑤 = −1 𝑤 = 2
◼ A topological defect is a group of spins that have a different topology
than spins that point only one direction
◼ A vortex is a special type of topological defect by having non-zero
winding number
➢ A spin configuration with defects cannot be smoothly transformed into the ferromagnetic
ground state where all spins are aligned
Vortex Anti-vortex Vortex
18/30

19. XY model –Topological Order
◼ Observations (𝑙th-sample): 𝑆𝑖
(𝑙)
= cos 𝜃𝑖
(𝑙)
, sin 𝜃𝑖
(𝑙)
𝜌 𝜃𝑖 ∝ 𝑒−𝐸({𝜃𝑖})/𝑇
𝐸 𝜃𝑖 = −𝐽 ෍
𝑖,𝑗
cos(𝜃𝑖 − 𝜃𝑗)
◼ Initialize topological defects at 𝑇 = 0
For 1D-XY model
𝜃𝑖
(𝑙)
= 2𝜋𝜈(𝑙)
𝑖
𝑁
+ 𝛿𝜃𝑖
(𝑙)
+ ҧ
𝜃(𝑙)
Winding
number
Spin
fluctuation
Global
rotation
For 2D-XY model
𝜃(𝑖,𝑗)
(𝑙)
= 2𝜋𝜈𝑥
(𝑙) 𝑖
𝑁𝑥
+ 2𝜋𝜈𝑦
(𝑙) 𝑗
𝑁𝑦
+ 𝛿𝜃(𝑖,𝑗)
(𝑙)
+ ҧ
𝜃(𝑙)
Winding number (𝑣𝑥, 𝑣𝑦)
(Rodriguez-Nieva+, Nat. Phys., 2019)
19/30

20. 2D XY model –Topological Order
◼ Initial Purpose: use persistent
homology to identify topological
sectors from spins configuration
𝑣𝑥, 𝑣𝑦 = 0, 0 , 0, 2 , 1, 1 , (2, −1)
𝑁 × 𝑁 spins, 𝑚𝜈 = 100 samples per 𝑣𝑥, 𝑣𝑦
Total = 400 samples at each temperature
◼ Use the Metropolis algorithm to
thermalize samples to temperatureT
(many thanks to J. F. Rodriguez-Nieva for his instruction)
Topological
Persistence Machine
𝑣𝑥, 𝑣𝑦 = ?
20/30

21. 2D XY model –Topological Order
𝑑 𝑠𝑖, 𝑠𝑗 = 𝜉𝑑 𝑺𝑖, 𝑺𝑗 + 1.0 − 𝜉 𝑑 𝒓𝑖, 𝒓𝑗
= 𝜉 |𝜃𝑖 − 𝜃𝑗| + (1.0 − 𝜉) 𝒓𝑖 − 𝒓𝑗
◼ Embedding: we define the distance between spin 𝑖-
th index and 𝑗-th index in 𝑁 × 𝑁 lattice as
Angle distance Lattice distance
◼ Compute Persistence Diagrams of loops: 𝐷𝑙
(𝑇)
for sample 𝑙-th (𝑙 = 1,2, … , 400)
◼ Compute Gram Matrix {𝑘𝑙𝑙′} from Kernel 𝑘 𝐷𝑙
𝑇
, 𝐷𝑙′
𝑇
at eachT
◼ Dimensional Reduction from Gram matrix to see the difference in the
topological order
21/30

22. 2D XY model –Topological Order
Results from Kernel PCA
𝑣𝑥, 𝑣𝑦 =
Distinguishable
Topological sector by winding number
Indistinguishable =Topological Order is lost
We can perform an unsupervised learning of the topological phase transition by
detecting the value of 𝑇 that fails to identify the topological sectors.
What can the Persistence Diagrams tell us more?
22/30

23. 2D XY model –Topological PhaseTransition
𝑣𝑥, 𝑣𝑦 = −1, 2 𝑚𝜈 = 10 samples per 𝑣𝑥, 𝑣𝑦 at each T
𝑇 = {0.30,0.31, … , 1.50} ,Total = 1210 samples
We focus on only one topological sector and see the varying of diagrams via
temperature
Group of well-ordered spins
Group of spins that form
vortices or antivortices
At high 𝑇, it is easy for vortices and antivortices to
appear in many places in the spin configuration
23/30

24. 2D XY model –Topological PhaseTransition
24/24
Dimensional Reduction from Gram
matrix using UMAP (McInnes+,2018)
Kernel Spectral
Clustering
𝑚𝜈 = 10 samples per 𝑣𝑥, 𝑣𝑦 at each T
𝑇
𝐽
≈ 0.89
The transition in the
proportion of diagrams
belonging to each cluster
The number of diagrams
grouped into the cluster
of the low-temperature regime
Q. H.Tran et al., Topological persistence machine of phase
transition, PRE (2021)
24/30

25. Case-study: Quantum Many Body
𝐻𝐼 = −𝐽 ෍
𝑖=1
𝐿−1
ො
𝜎𝑖
𝑧
ො
𝜎𝑖+1
𝑧
− 𝐽𝑔 ෍
𝑖=1
𝐿
ො
𝜎𝑖
𝑥 𝐻𝐵 = −𝑡 ෍
𝑖=1
𝐿−1
෠
𝑏𝑖
† ෠
𝑏𝑖+1 + ෠
𝑏𝑖+1
† ෠
𝑏𝑖
One-dimensional Transverse Ising
model
One-dimensional Bose Hubbard model
ො
𝑛𝑖 = ෠
𝑏𝑖
† ෠
𝑏𝑖
+
𝑈
2
෍
𝑖=1
𝐿
ො
𝑛𝑖 ො
𝑛𝑖 − 𝕝 − 𝜇 ෍
𝑖=1
𝐿
ො
𝑛𝑖
𝑔𝑐 = 1.0 𝑔
𝑇
𝑇 = 0
Domain-wall
quasiparticles
Flipped-spin
quasiparticles
Ordered phase
(𝑇 = 0) QPT at ground state
Quantum
Critical 𝑇 = 0
L. D. Carr et al., Mesoscopic effects in quantum phases of
ultracold quantum gases in optical lattices, PRA (2010)
𝐿 = 51
25/30

26. Quantum Many Body
◼ At ground state ො
𝜌, we define the quantum mutual information matrix
ℒ𝑖𝑗 =
1
2
𝑆𝑖 + 𝑆𝑗 − 𝑆𝑖𝑗 for 𝑖 ≠ 𝑗; ℒ𝑖𝑖 = 0
𝑆𝑖 = 𝑇𝑟 ො
𝜌𝑖 log ො
𝜌𝑖
◼ Consider {ℳ𝑖𝑗} as a weighted-graph
𝑑(𝑖, 𝑗) = 1 − 𝑟𝑖𝑗
2
◼ Define the distance where 𝑟𝑖𝑗 is the Pearson
correlation coefficient
𝑆𝑖𝑗 = 𝑇𝑟[ො
𝜌𝑖𝑗 log ො
𝜌𝑖𝑗]
ො
𝜌𝑖 = 𝑇𝑟𝑘≠𝑖 ො
𝜌
ො
𝜌𝑖𝑗 = 𝑇𝑟𝑘≠𝑖,𝑗 ො
𝜌
◼ Observations from the quantum many body system may not have explicit shapes
M.A.Valdez et al., Quantifying Complexity in Quantum PhaseTransitions
via Mutual Information Complex Networks, PRL (2017)
26/30

27. QPT – IsingTransverse
𝒫𝛾 𝐷 = ෍
𝑏,𝑑 ∈𝐷
𝑑 − 𝑏 𝛾
1/𝛾
ℰ 𝐷 = −
1
log 𝒫1(𝐷)
෍
𝑏,𝑑 ∈𝐷
|𝑑 − 𝑏|
𝒫1(𝐷)
log
|𝑑 − 𝑏|
𝒫1(𝐷)
Persistence Diagrams
calculated for connected
components
Death − Birth
Q. H.Tran et al., Topological persistence
machine of phase transition, PRE (2021)
27/30

28. QPT Bose-Hubbard
For loops
◼ Fitting 𝑦 𝐿 = 𝑦(∞) + 𝛼𝐿−𝛽
for 𝐿 → ∞
𝑡
𝑈 𝐵𝐾𝑇
= 0.289 ± 0.001
Our
Density-matrix renormalization
𝑡
𝑈 𝐵𝐾𝑇
= 0.29 ± 0.01
Death − Birth
◼ For small 𝐿:
𝑦 𝐿 =
𝑡
𝑈 𝐵𝐾𝑇
≈ 0.2
Q. H.Tran et al., Topological persistence machine of
phase transition, PRE (2021)
28/30
T. D. Kühner et al., One-dimensional
Bose-Hubbard model with nearest-neighbor
interaction, PRB (2000).
For connected components

29. Summary
◼ We apply persistent homology for the raw data of physical states to
identify the phase of matter with appropriate interpretation.
29/30
◼ Without prior knowledge, this approach provides potential in
general system where the Hamiltonian may be unknown.
◼ The indicator from persistent homology only represents a necessary
but not sufficient condition.
◼ It would be interesting if we can come up this approach for “model
explainability”.

30. Thank You
and Wish
You all the
Best!